{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-06T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":51322,"title":"Solve an ODE: diffusion problem 1","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 264.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.125px; transform-origin: 407px 132.125px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a eta f' = 0\" style=\"width: 91px; height: 18px;\" width=\"91\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.1167px 7.91667px; transform-origin: 45.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHgAAAAoCAYAAAA16j4lAAAD1UlEQVR4nO1af5HzIBB9HuKgBmKgCk5BHcRBHdRCNVRCPdRCNNRCvz/gTfbypbsLgaa54c1kbpojLPD2JwA0NDQ0NDQ0NDR8NQ4AjonfdACu8W+pMVwK9veN6OPzUYEjgDsCWU8Ag+O7Ln5jDfYA4ATgHB+r/RHAA3+P5BPC2l4R1m1EukEl4wjgBeAWfw/x9xP6AncIJGgDpHW/Yr9HTJO8QSf6FPv/KzhjWgcgzP+FQHQ1dAiL/USwMgD4cQq+IbhSre9H7Os0+x+VaoRO8tWQsRdwvlJhL/HduaZgatXNajgDrVCzcE7gnRXSsjXZVMDqbqwy7vhtvR/DE8sWZmGErnmH2K+moT+izY/S1yXK2yt6TPM8GG2L4pQpmN95rPeF99bXiTZXpS8qi6YE3wx6qo8rKYN8quAH7Pg8YiJPUwS2ezpkpoaRbwG9pKbExTBgKlUoeBTvBuiE0JqsxIDkWsTdRVst2aJH2AM6TOspPdldvE8NiW5QAN0GkxyvYLpnLelhxphKsCbbI1cbz7nA4w1jB/GNnN9FvK+eNDJ7Tl00fqdZmyTYcv830VbLMI+ONu8g57rmydl5IsFWxVEcXNhUwRywBkmwFTfl4mvkMRPNqRlLWXAOQQyDVTczSgpOJdjaiZKhwrLOnHp9S8jyqOpmxhyyRk3dJfIQLCdWKgYDG1nCCsgy9F2Jd8a0I/hQ2iXBu8GwBA/BQB7BVozbG8Eyg15y7zxsYPJGw1i922UJ1uAlmHvQVoz3tvOWZ0v4dBZNcG5LYYrWPZ8Pc6NVCRlJyjmpoXJYA5BKpFmmrBE1MK7nELxVFq2FQZI/96Ac66qYTcE5Oys8SrRKq5J70bLtlnVwilXJRHMpt3jnQZdOnZIgE6Cc3ZSUOOE9TfLEVWr2Xi4A0BBe+N+1S/LfEewJg8mCvRjhs37tPJiKIs+hNTywr8N/bZ/fS3CWMpc42bjAzo4JXumh1csbHTKDtPrI9ThbgYcoS4ZQlWBa1JpbEoyvKQveYzroGJCWsAzYYKtvBeQx6FJuIcNkUYKl4LU3+i74XE06YoObECvAhFDzclWSLNZeJWJZh7Dwtd3mGfuKvcAUBjUvybA1rwqYTJoelrcZpeYz8Jc6quJV21rXUHr4k7Ct0COss3TFvMSouVhro8P0sLK47zD5/dK3Cn4QtLF0fGQG/tHL4RmYbxqxSvF4tjt+KwI5cuVH1H66UW9pk4MeZUnuYN+Z/hbQEu8I5KbkC7z58cB0OSAp5B2isAH13VxJC+4K91cbPabbMHsad0NDQ0NDQ0NDQ0NDw17xDxRpzXmzha10AAAAAElFTkSuQmCC\" alt=\"f(0) = f0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(infinity) = 0\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.133px 7.91667px; transform-origin: 372.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion1ODE(eta,a,f0)\r\n%  eta = independent variable\r\n%  a   = constant\r\n%  f0  = value of f at eta = 0\r\n\r\n   f = g(eta,a,f0)\r\nend","test_suite":"%%\r\na = 1/2; \r\nf0 = 1;\r\neta = 0:0.2:1;\r\nf_correct = [1 0.887537083981715 0.777297410789522 0.671373240540873 0.571607644953331 0.479500122186953];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2; \r\nf0 = 1;\r\neta = 0.5;\r\nf_correct = 0.479500122186953;\r\nassert(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\nf0 = 0.5;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.452241573979416 0.409045884857994 0.324194989107724 0.215056003266342 0.156008214896009];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2;\r\nf0 = 1;\r\neta = 1:4;\r\nf_correct = [0.157299207050285 0.004677734981047 2.209049699858544e-05 1.541725790028002e-08];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 3/4;\r\nf0 = 4/3;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [1.305696910508165 1.060016229285651 0.012499691279247];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 0.01;\r\nf0 = 1;\r\neta = rand/120;\r\nf_correct = polyval(flip([1 -0.0797885 0 0.000132981 0 -1.99471e-7]),eta);\r\nf = diffusion1ODE(eta,a,f0);\r\nassert(all(abs(f-f_correct)\u003c1e-7))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-08T01:02:26.000Z","updated_at":"2025-05-04T20:55:44.000Z","published_at":"2021-04-08T01:09:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a eta f' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime + a\\\\eta f\\\\prime = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(infinity) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51625,"title":"Solve an ODE: diffusion problem 2","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 317px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 158.5px; transform-origin: 407px 158.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 7.91667px; transform-origin: 63.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a(f+eta f') = 0\" style=\"width: 130px; height: 19px;\" width=\"130\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.3333px 7.91667px; transform-origin: 44.3333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral(f,{eta,-inf,inf}) = 1\" style=\"width: 78px; height: 44px;\" width=\"78\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 349.967px 7.91667px; transform-origin: 349.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion2ODE(eta,a)\r\n%  eta = independent variable\r\n%  a   = positive constant\r\n  f = a*sqrt(eta);\r\nend","test_suite":"%%\r\na = 1/2; \r\neta = 0:0.2:1;\r\nf_correct = [0.282094791773878 0.279287901697234 0.271033696776216 0.257815227404741 0.240385324709827 0.219695644733861];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2; \r\neta = 0.5;\r\nf_correct = 0.439391289467722;\r\nassert(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13)\r\n\r\n%%\r\na = 1;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.396080211793656 0.388528585315836 0.359548380672770 0.292234407541508 0.239309153606614];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2;\r\neta = 1:4;\r\nf_correct = [0.207553748710297 0.010333492677046 0.000069626525973 0.000000063491173];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 3/4;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [0.345377564870647 0.334028296534584 0.011822159682599];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 1;\r\neta = rand/120;\r\nf_correct = polyval([1/10321920 0 -1/645120 1/46080 0 -1/3840 0 1/384 0 -1/48 0 1/8 0 -1/2 0 1]./(sqrt(2*pi)),eta);\r\nf = diffusion2ODE(eta,a);\r\nassert(all(abs(f-f_correct)\u003c1e-13))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-01T14:31:00.000Z","updated_at":"2021-05-01T14:34:52.000Z","published_at":"2021-05-01T14:34:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a(f+eta f') = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime+a(f+\\\\eta f\\\\prime)=0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral(f,{eta,-inf,inf}) = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty fd\\\\eta=1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":51322,"title":"Solve an ODE: diffusion problem 1","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 264.25px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.125px; transform-origin: 407px 132.125px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 7.91667px; transform-origin: 28px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a eta f' = 0\" style=\"width: 91px; height: 18px;\" width=\"91\" height=\"18\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.25px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.125px; text-align: left; transform-origin: 384px 21.125px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.1167px 7.91667px; transform-origin: 45.1167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The problem has the conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(0) = f0\" style=\"width: 60px; height: 20px;\" width=\"60\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f(infinity) = 0\" style=\"width: 65px; height: 19px;\" width=\"65\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 372.133px 7.91667px; transform-origin: 372.133px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion1ODE(eta,a,f0)\r\n%  eta = independent variable\r\n%  a   = constant\r\n%  f0  = value of f at eta = 0\r\n\r\n   f = g(eta,a,f0)\r\nend","test_suite":"%%\r\na = 1/2; \r\nf0 = 1;\r\neta = 0:0.2:1;\r\nf_correct = [1 0.887537083981715 0.777297410789522 0.671373240540873 0.571607644953331 0.479500122186953];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2; \r\nf0 = 1;\r\neta = 0.5;\r\nf_correct = 0.479500122186953;\r\nassert(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10)\r\n\r\n%%\r\na = 1;\r\nf0 = 0.5;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.452241573979416 0.409045884857994 0.324194989107724 0.215056003266342 0.156008214896009];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 2;\r\nf0 = 1;\r\neta = 1:4;\r\nf_correct = [0.157299207050285 0.004677734981047 2.209049699858544e-05 1.541725790028002e-08];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 3/4;\r\nf0 = 4/3;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [1.305696910508165 1.060016229285651 0.012499691279247];\r\nassert(all(abs(diffusion1ODE(eta,a,f0)-f_correct)\u003c1e-10))\r\n\r\n%%\r\na = 0.01;\r\nf0 = 1;\r\neta = rand/120;\r\nf_correct = polyval(flip([1 -0.0797885 0 0.000132981 0 -1.99471e-7]),eta);\r\nf = diffusion1ODE(eta,a,f0);\r\nassert(all(abs(f-f_correct)\u003c1e-7))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-04-08T01:02:26.000Z","updated_at":"2025-05-04T20:55:44.000Z","published_at":"2021-04-08T01:09:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a eta f' = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime + a\\\\eta f\\\\prime = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The problem has the conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(0) = f0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(0) = f_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f(infinity) = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(\\\\infty ) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity from one point where the concentration is maintained at a constant value into a medium in which the concentration is zero far away. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":51625,"title":"Solve an ODE: diffusion problem 2","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 317px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 158.5px; transform-origin: 407px 158.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 7.91667px; transform-origin: 63.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 295.017px 7.91667px; transform-origin: 295.017px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"f''+a(f+eta f') = 0\" style=\"width: 130px; height: 19px;\" width=\"130\" height=\"19\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.91667px; transform-origin: 21.0083px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ewhere \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 298.233px 7.91667px; transform-origin: 298.233px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 44.3333px 7.91667px; transform-origin: 44.3333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"integral(f,{eta,-inf,inf}) = 1\" style=\"width: 78px; height: 44px;\" width=\"78\" height=\"44\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 194.342px 7.91667px; transform-origin: 194.342px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to solve this problem—that is, return values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 69.2333px 7.91667px; transform-origin: 69.2333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e at specified values of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eη\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8333px 7.91667px; transform-origin: 40.8333px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBackground\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 349.967px 7.91667px; transform-origin: 349.967px 7.91667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = diffusion2ODE(eta,a)\r\n%  eta = independent variable\r\n%  a   = positive constant\r\n  f = a*sqrt(eta);\r\nend","test_suite":"%%\r\na = 1/2; \r\neta = 0:0.2:1;\r\nf_correct = [0.282094791773878 0.279287901697234 0.271033696776216 0.257815227404741 0.240385324709827 0.219695644733861];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2; \r\neta = 0.5;\r\nf_correct = 0.439391289467722;\r\nassert(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13)\r\n\r\n%%\r\na = 1;\r\neta = [0.12 0.23 0.456 0.789 1.011];\r\nf_correct = [0.396080211793656 0.388528585315836 0.359548380672770 0.292234407541508 0.239309153606614];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 2;\r\neta = 1:4;\r\nf_correct = [0.207553748710297 0.010333492677046 0.000069626525973 0.000000063491173];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 3/4;\r\neta = 3*logspace(-2,0,3);\r\nf_correct = [0.345377564870647 0.334028296534584 0.011822159682599];\r\nassert(all(abs(diffusion2ODE(eta,a)-f_correct)\u003c1e-13))\r\n\r\n%%\r\na = 1;\r\neta = rand/120;\r\nf_correct = polyval([1/10321920 0 -1/645120 1/46080 0 -1/3840 0 1/384 0 -1/48 0 1/8 0 -1/2 0 1]./(sqrt(2*pi)),eta);\r\nf = diffusion2ODE(eta,a);\r\nassert(all(abs(f-f_correct)\u003c1e-13))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-05-01T14:31:00.000Z","updated_at":"2021-05-01T14:34:52.000Z","published_at":"2021-05-01T14:34:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the solution of a problem involving diffusion, the following ordinary differential equation arises\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f''+a(f+eta f') = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\\\\prime\\\\prime+a(f+\\\\eta f\\\\prime)=0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"a\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a positive constant and primes denote differentiation with respect to the independent variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and its derivatives vanish at infinity, and it is subject to the constraint\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"integral(f,{eta,-inf,inf}) = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\int_{-\\\\infty}^\\\\infty fd\\\\eta=1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to solve this problem—that is, return values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"f\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e at specified values of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"eta\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\eta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBackground\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe physical problem involves diffusion of a quantity instantaneously injected at a point—for example, a spill of a contaminant in an initially clean river. The ODE results from transforming the diffusion equation—a partial differential equation in time and a spatial coordinate—with a similarity solution. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"tag:\"diffusion\"","current_player_id":null,"fields":[{"name":"page","type":"integer","callback":null,"default":1,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"per_page","type":"integer","callback":null,"default":50,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"sort","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"body","type":"text","callback":null,"default":"*:*","directive":null,"facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":false},{"name":"group","type":"string","callback":null,"default":null,"directive":"group","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"difficulty_rating_bin","type":"string","callback":null,"default":null,"directive":"difficulty_rating_bin","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"id","type":"integer","callback":null,"default":null,"directive":"id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"tag","type":"string","callback":null,"default":null,"directive":"tag","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"product","type":"string","callback":null,"default":null,"directive":"product","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_at","type":"timeframe","callback":{},"default":null,"directive":"created_at","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"profile_id","type":"integer","callback":null,"default":null,"directive":"author_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_by","type":"string","callback":null,"default":null,"directive":"author","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player_id","type":"integer","callback":null,"default":null,"directive":"solver_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player","type":"string","callback":null,"default":null,"directive":"solver","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"solvers_count","type":"integer","callback":null,"default":null,"directive":"solvers_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"comments_count","type":"integer","callback":null,"default":null,"directive":"comments_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"likes_count","type":"integer","callback":null,"default":null,"directive":"likes_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leader_id","type":"integer","callback":null,"default":null,"directive":"leader_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leading_solution","type":"integer","callback":null,"default":null,"directive":"leading_solution","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true}],"filters":[{"name":"asset_type","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":"\"cody:problem\"","prepend":true},{"name":"profile_id","type":"integer","callback":{},"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":"author_id","static":null,"prepend":true}],"query":{"params":{"per_page":50,"term":"tag:\"diffusion\"","current_player":null,"sort":"map(difficulty_value,0,0,999) asc"},"parser":"MathWorks::Search::Solr::QueryParser","directives":{"term":{"directives":{"tag":[["tag:\"diffusion\"","","\"","diffusion","\""]]}}},"facets":{"#\u003cMathWorks::Search::Field:0x00007faf50c41e20\u003e":null,"#\u003cMathWorks::Search::Field:0x00007faf50c41d80\u003e":null},"filters":{"#\u003cMathWorks::Search::Field:0x00007faf50c41420\u003e":"\"cody:problem\""},"fields":{"#\u003cMathWorks::Search::Field:0x00007faf50c420a0\u003e":1,"#\u003cMathWorks::Search::Field:0x00007faf50c42000\u003e":50,"#\u003cMathWorks::Search::Field:0x00007faf50c41f60\u003e":"map(difficulty_value,0,0,999) asc","#\u003cMathWorks::Search::Field:0x00007faf50c41ec0\u003e":"tag:\"diffusion\""},"user_query":{"#\u003cMathWorks::Search::Field:0x00007faf50c41ec0\u003e":"tag:\"diffusion\""},"queried_facets":{}},"query_backend":{"connection":{"configuration":{"index_url":"http://index-op-v2/solr/","query_url":"http://search-op-v2/solr/","direct_access_index_urls":["http://index-op-v2/solr/"],"direct_access_query_urls":["http://search-op-v2/solr/"],"timeout":10,"vhost":"search","exchange":"search.topic","heartbeat":30,"pre_index_mode":false,"host":"rabbitmq-eks","port":5672,"username":"cody-search","password":"78X075ddcV44","virtual_host":"search","indexer":"amqp","http_logging":"true","core":"cody"},"query_connection":{"uri":"http://search-op-v2/solr/cody/","proxy":null,"connection":{"parallel_manager":null,"headers":{"User-Agent":"Faraday v1.0.1"},"params":{},"options":{"params_encoder":"Faraday::FlatParamsEncoder","proxy":null,"bind":null,"timeout":null,"open_timeout":null,"read_timeout":null,"write_timeout":null,"boundary":null,"oauth":null,"context":null,"on_data":null},"ssl":{"verify":true,"ca_file":null,"ca_path":null,"verify_mode":null,"cert_store":null,"client_cert":null,"client_key":null,"certificate":null,"private_key":null,"verify_depth":null,"version":null,"min_version":null,"max_version":null},"default_parallel_manager":null,"builder":{"adapter":{"name":"Faraday::Adapter::NetHttp","args":[],"block":null},"handlers":[{"name":"Faraday::Response::RaiseError","args":[],"block":null}],"app":{"app":{"ssl_cert_store":{"verify_callback":null,"error":null,"error_string":null,"chain":null,"time":null},"app":{},"connection_options":{},"config_block":null}}},"url_prefix":"http://search-op-v2/solr/cody/","manual_proxy":false,"proxy":null},"update_format":"RSolr::JSON::Generator","update_path":"update","options":{"url":"http://search-op-v2/solr/cody"}}},"query":{"params":{"per_page":50,"term":"tag:\"diffusion\"","current_player":null,"sort":"map(difficulty_value,0,0,999) asc"},"parser":"MathWorks::Search::Solr::QueryParser","directives":{"term":{"directives":{"tag":[["tag:\"diffusion\"","","\"","diffusion","\""]]}}},"facets":{"#\u003cMathWorks::Search::Field:0x00007faf50c41e20\u003e":null,"#\u003cMathWorks::Search::Field:0x00007faf50c41d80\u003e":null},"filters":{"#\u003cMathWorks::Search::Field:0x00007faf50c41420\u003e":"\"cody:problem\""},"fields":{"#\u003cMathWorks::Search::Field:0x00007faf50c420a0\u003e":1,"#\u003cMathWorks::Search::Field:0x00007faf50c42000\u003e":50,"#\u003cMathWorks::Search::Field:0x00007faf50c41f60\u003e":"map(difficulty_value,0,0,999) asc","#\u003cMathWorks::Search::Field:0x00007faf50c41ec0\u003e":"tag:\"diffusion\""},"user_query":{"#\u003cMathWorks::Search::Field:0x00007faf50c41ec0\u003e":"tag:\"diffusion\""},"queried_facets":{}},"options":{"fields":["id","difficulty_rating"]},"join":" "},"results":[{"id":51322,"difficulty_rating":"easy-medium"},{"id":51625,"difficulty_rating":"easy-medium"}]}}